Unable to align extruded text to grid bottom
You could set a driver to Z Location:

However, this can cause glitching in Edit Mode:

You could therefore use a Python script to align objects whenever you need - go to scripting tab, create a new file, paste the below and press the Play  button to realign:
import bpy

aligned_objects = ['Cube']

for name in aligned_objects:
    aligned_object = bpy.data.objects[name]
    aligned_object.location.z = - aligned_object.bound_box[0][2]